Method and program for converting boundary data into cell inner shape data

ABSTRACT

A method and a program for converting boundary data into cell inner shape data, includes a division step (A) of dividing external data ( 12 ) constituted of the boundary data of an object into cells ( 13 ) in an orthogonal grid, a cutting point deciding step (B) of deciding an intersection point of the boundary data and a cell edge as a cell edge cutting point, a boundary deciding step (C) of deciding a boundary formed by connecting the cell edge cutting points as the cell inner shape data, a cell classification step (D) of classifying the divided cells into a nonboundary cell ( 13   a ) including no boundary surface and a boundary cell ( 13   b ) including a boundary surface, and a boundary cell data classification step (E) of classifying cell data constituting the boundary cell into internal cell data inside the cell inner shape data and external cell data outside the cell inner shape data.

This is a National Phase Application in the United States ofInternational Patent Application No. PCT/JP03/02197 filed Feb. 27, 2003,which claims priority on Japanese Patent Application No. 053575/2002,filed Feb. 28, 2002. The entire disclosures of the above patentapplications are hereby incorporated by reference.

BACKGROUND OF THE INVENTION

1. Technical Field of the Invention

The present invention relates to a method for storing substantial datawhich can unify CAD and simulation by storing the substantial data thatintegrates a shape and physical properties by a small storage capacity,and more particularly to a method and a program for converting boundarydata into cell inner shape data.

2. Description of the Related Art

In a field of advanced research and development/technical developments,a higher level/complexity thereof has made a great many trials anderrors absolutely necessary, increasing risks in the middle ofdevelopments. In Japan that depends on science and technology for itssurvival, it is extremely important to achieve an unprecedentedly highlevel/efficiency of a development process by eliminating such risks asmany as possible.

In the field of research and development/technical developments,computer aided design (CAD), computer aided manufacturing (CAM),computer aided engineering (CAE), computer aided testing (CAT), and thelike are currently used as simulation means of designing, fabricating,analyzing and testing.

Because of the present invention, it is expected that cooperativesimulation (C-Simulation) which is continuous simulation, advanced CAM(A-CAM) which takes a fabrication process into consideration,deterministic fabrication (D-fabrication) which can achieve ultimateaccuracy, and the like will come into wide use.

According to such conventional simulation means, data of an object isstored based on constructive solid geometry (CSG) or boundaryrepresentation (B-rep).

In the case of the CSG, however, the entire object is stored as anaggregation of very small solid models. Consequently, if data is heavyand simulation means (software or the like) is mounted, enormous datamust be processed, causing a problem of much time necessary for analysiseven when a large scale computer is used.

In the case of the B-rep, the object is represented by a boundary. Thus,while data is light and an amount of data is small, there is not directinformation regarding the inside of a boundary surface, causing aproblem of unsuitability to deformation analysis or the like if nochange is made.

Furthermore, according to the conventional data storage means, each timethermal/fluid analysis, large solid analysis, coupled analysis thereofor the like is carried out, division is made in a mesh form or the likesuited to the analysis, and a result of the analysis can be displayed orthe like to apply a finite element method or the like. However,unification of CAD and simulation is difficult, causing a problem ofimpossibility of managing the processes of designing, analyzing,fabricating, assembling, testing and the like based on the same data.

In other words, the following problems are inherent in the currentsolid/surface-CAD (referred to as S-CAD hereinafter):

-   (1) data is not passed, inferior in internal conversion operation    (problems of numerical value error and processing method);-   (2) direct use is impossible for simulation (mesh must be formed);    and-   (3) investigation of fabrication by CAM is impossible (only last    shape is given).

Additionally, the following problems are inherent in fabrication:

-   (1) a fabrication process cannot be represented (rough fabrication    or process design assistance is insufficient);-   (2) dealing with a new fabrication method such as laser fabrication    or superadvanced fabrication is impossible (only cutting is    available, numerical value accuracy is insufficient); and-   (3) a fabrication method itself cannot be selected (different    material characteristics are given in compound material).

To solve the aforementioned problems, the inventors et. al have invented“METHOD FOR STORING SUBSTANTIAL DATA THAT INTEGRATES SHAPE AND PHYSICALPROPERTIES”, and applied for a patent (Japanese Patent Application No.2001-25023, not laid-open).

According to this invention, as schematically shown in FIG. 1, externaldata constituted of boundary data of an object is divided into cubiccells by oct-tree division in which boundary surfaces cross each other,and the cells are classified into a nonboundary cell 13 a which includesno boundary surface and a boundary cell 13 b which includes a boundarysurface. In the drawing, a reference numeral 15 is a cutting point.

According to this invention, various physical property values are storedfor each cell, and substantial data that integrates shapes and physicalproperties can be stored by a small storage capacity. Thus, a shape, astructure, physical property information, and hysteresis are managed ina unified manner to enable management of data regarding a series ofprocesses from designing to fabricating, assembling, testing, evaluationand the like based on the same data, whereby it is possible to unify CADand simulation.

The aforementioned method for storing the substantial data is referredto as “volume CAD” or “V-CAD” hereinafter. In the present application,the V-CAD is defined as follows: “V-CAD means that a boundary surface isformed in a cell of a voxel dataset”.

According to conventional CAD, even a solid is in fact hollowpapier-mache stage data as in the case of the B-rep or the like. On theother hand, according to the V-CAD, even the inside is stuffed, andphysical data can be held. Because of internal information that has beenprovided, it is expected that geometrical calculation which tends tobreak down in shape processing of the B-rep or the like can be stronglycarried out. Further, the V-CAD goes beyond a framework of a simple toolto represent shapes, and is designed to provide a data foundation whichcan be directly used for simulation and fabrication. In order to trulyachieve a system of such “manufacturing”, a simulation technology or afabrication technology must be simultaneously developed to effectivelyuse the V-CAD. Especially, for fabrication, only data of a surface shapehas been used. Therefore, it can be said that there are almost nofabrication technologies capable of truly utilizing volume data exceptlaser stereolithography and rapid prototyping (3D ink jet).

From the viewpoint of the current situation of a manufacturing world, itis very important to generate volume data in the V-CAD by reading ashape represented by the conventional type CAD. Thus, according to theV-CAD, it is necessary to possess boundary data which enablesreconstruction of a boundary of shape data (external data) in a boundarycell.

Conventionally, it is marching cubes (abbreviated to MC hereinafter)that have generally been used for generating a polygon from volume data.For example, the MC is introduced in the following Document 1:

(Document 1) “MARCHING CUBES: A HIGH RESOLUTION 3D SURFACE CONSTRUCTIONALGORITHM”, Computer Graphics, Volume 21, Number 4, July 1987.

For reference, FIGS. 2 a to 2 d show all cutting point patterns andboundary segments of two-dimensional MC, and FIGS. 3 a to 3 n show allcutting point patterns (boundary surfaces are omitted) ofthree-dimensional MC.

In the case of the three-dimensional MC, positive and negative numericalvalues are written in 8 vertexes of a three-dimensional cell (cube), andisosurfaces are generated based on these numerical values (isosurfacesof zero values are considered hereinafter). One cutting point isdisposed on an edge if signs (positive or negative) of numerical valuesof both ends of the edge of the cube do not match each other. No cuttingpoint is disposed if they match each other. This operation is carriedout for 12 edges of the cube, and then planes are formed based oncutting points. The same holds true for the two-dimensional MC.

FIGS. 4 a to 4 c are exemplary views showing a difference of cuttingpoints in rectangular cells between MC and Kitta cubes (KTC). In theexamples, in the case of the MC, 4 vertexes (white circles) of a squarecell have the same sign as they are located outside a shape (closedcurve) in a situation as shown in FIG. 4 a, and thus no cutting pointsare generated on 4 edges (4 sides) of the cell. As a result, noapproximate isosurfaces are formed at all in this case. This means thatcurrent resolution is too rough to represent the shape from thestandpoint of the MC. Therefore, in the MC of the example, there is aproblem of impossibility of representing the cutting points of the 4edges of the cell as in the case of FIG. 4 b or 4 c. FIG. 4 a shows anextreme example. Essentially similar defects frequently occur, and FIG.4 d shows an defect example. Such defects frequently occur at anintersection of a curved boundary surface and a cell edge. In the caseof the KTC, this situation is approximated as shown in FIG. 4 f. In thecase of the MC, it is approximated as shown in FIG. 4 g. FIGS. 4 c and 4f show two-dimensional examples. In three-dimensional representation,more cases can be represented only by the KTC. It can be understood thatthe KTC has much richer power of representation than the MC at equalresolution.

On the other hand, according to the present invention (KTC) describedlater, two cutting points are generated on each of 4 edges as shown inFIG. 4 b. If the number of cutting points is limited to 0 or 1 on oneedge, representation as shown in FIG. 4 c can be obtained.

FIGS. 5 a and 5 b are views showing a difference of cutting points onedges between the conventional MC and the KTC of the present invention.When the MC is constructed, the number of cutting points is limited to 0or 1 on one edge. As illustrated in FIG. 5 a, in the case of the MC, onecutting point is generated on an edge only when signs of both ends ofeach edge of a cell oppose each other (positive and negative values).Thus, as shown in FIG. 5 b, when a cutting point is given on one edge,signs of both edges may not only oppose each other (positive andnegative values) but also match each other. In the case of the MC, acutting point can be represented only in a part of such cases.

In FIGS. 4 a to 4 c, in the case of the MC, the cell must be subdividedto represent a shape indicated by a closed solid line. As a result, inthe V-CAD that uses the MC, subdivision of the cell becomes necessary tohold boundary data, and a storage capacity to store substantial datathat integrates the shape and physical properties is accordinglyincreased exponentially. Furthermore, prevention of an increase in thestorage capacity causes a difficulty of precisely representing a shapeof a boundary part.

SUMMARY OF THE INVENTION

The present invention has been made to solve the foregoing problems.That is, objects of the invention are to provide a method and a programfor converting boundary data into cell inner shape data, which caninclude all cell edge cutting points obtained by marching cubes (MC),and all cell edge cutting points unobtainable by the MC withoutomission, and thereby include cell inner shape data constituted ofboundaries that connect the cell edge cutting points without omission.

According to the present invention, there are provided a method and aprogram for converting boundary data into cell inner shape data,characterized by comprising: a division step (A) of dividing externaldata (12) constituted of boundary data of an object into cells (13) inan orthogonal grid; a cutting point deciding step (B) of deciding anintersection point of the boundary data and a cell edge as a cell edgecutting point; a boundary deciding step (C) of deciding a boundaryconnecting formed by the cell edge cutting points as the cell innershape data; a cell classification step (D) of classifying the dividedcells into a nonboundary cell (13 a) including no boundary surface and aboundary cell (13 b) including a boundary surface; and a boundary celldata classification step (E) of classifying cell data constituting theboundary cell into internal cell data inside the cell inner shape dataand external cell data outside the cell inner shape data.

According to the method and the program, by the division step (A) andthe cell classification step (D), it is possible to store the externaldata (12) of the object as a cell hierarchy in which the external cell(12) is divided into the cells (13) in an orthogonal grid by a smallstorage capacity.

In the cutting point deciding step (B), the intersection point of theboundary data and the cell edge is decided as the cell edge cuttingpoint. Thus, it is possible to include arrangement of all cutting pointsby an MC in which “one cutting point is disposed on an edge if signs ofnumerical values of both ends of the cell edge are different from eachother, and no cutting point is disposed if signs of the numerical valuesare equal to each other”, and to include all other arrangements ofcutting points on boundary surfaces and cell edge lines without omissionunder a condition that the number of cutting points on one edge is atmost one.

Further, in the boundary deciding step (C), the boundary formed byconnecting the obtained cell edge cutting points is decided as the cellinner shape data. Thus, it is possible to include all cell inner shapepatterns by the MC, and to include other cell inner shape patternswithout omission under the condition that the number of cutting pointson one edge is at most one.

Additionally, in the boundary cell data classification step (E), thecell data constituting the boundary cell are classified into theinternal cell data inside the cell inner shape data and the externalcell data outside the cell inner shape data. Thus, it is possible toclassify all the cell data into nonboundary and boundary cell data whilemaintaining continuity from adjacent cells.

According to a preferred embodiment of the invention, the cells arequadrangular cells including square and rectangular cells intwo-dimensional representation, and in the cutting point deciding step(B), intersection points of boundary data and cell edges that havetotally 2⁴=16 arrangement cases (patterns or sorts) are decided as thecell edge cutting points, and the arrangement cases that becomeequivalence classes by rotational operation are decided as identicalpatterns on the assumption that the quadrangular cells are square cellsso that the 2⁴=16 arrangement cases are further classified into 6patterns.

By this method, in the case of the two-dimensional cell, it is possibleto classify patterns of cell edge cutting points into 6 patternsincluding 4 patterns by the MC, and to include all cell edge cuttingpoints possible without omission under the condition that the number ofcutting points on one edge is at most one.

In the boundary deciding step (C), arrangement of a cutting segment(=boundary line formed by connecting cell edge cutting points) isdecided as the cell inner shape data for all the 6 patterns.

Additionally, in the boundary deciding step (C), cell inner shape datapatterns that become equivalence classes by three-dimensional rotationaloperation are decided as identical patterns so that the cell inner shapedata patterns are classified into 22 patterns.

By these methods, it is possible to include all cell inner shape data bythe MC, and cell inner shape data possible under the condition that thenumber of cutting points on one edge is at most one, without omission.

According to another preferred embodiment of the invention, the cellsare hexahedron cells including cubic and rectangular parallelepipedcells, and in the cutting point deciding step (B), intersection pointsof boundary data and cell edges that have totally 2¹²=4096 arrangementcases are decided as the cell edge cutting points, and the arrangementcases that become equivalence classes by rotational operation andmirroring operation are decided as identical patterns so that the2¹²=4096 arrangement cases are further classified into 144 patterns.

By this method, in the case of the three-dimensional cell, it ispossible to classify patterns of cell edge cutting points into 144patterns including 14 patterns by the MC, and to include all otherpatterns of cell edge cutting points possible under the condition thatthe number of cutting points on one edge is at most one, withoutomission.

In the cutting point deciding step (B), the cell edge cutting pointpatterns that become equivalence classes by an inversion operationregarding presence/nonpresence of cutting points are decided asidentical patterns so that the cell edge cutting point patterns areclassified into 87 patterns in which the number of the cell edge cuttingpoints is 0 to 6.

By this method, it is possible to reproduce all the 144 patterns of thecell edge cutting points without omission by the 87 patterns in whichthe number of cell edge cutting points is 0 to 6.

In the boundary deciding step (C), a boundary surface formed byconnecting the cell edge cutting points is decided as the cell innershape data for all the patterns.

By this method, it is possible to include all the cell inner shape databy the MC, and all other cell inner shape data possible under thecondition that the number of cutting points on one edge is at most one,without omission.

Other objects and advantageous features of the present invention willbecome apparent upon reading of the following description with referenceto the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a principle view of V-CAD;

FIGS. 2 a to 2 d are views showing four kinds of equivalence classes of2D-MC;

FIGS. 3 a to 3 n are views showing all equivalence classes of cuttingpoint arrangement of 3D-MC;

FIGS. 4 a to 4 f are views of typical examples of KTC;

FIG. 4 g is a view showing representation of FIG. 4 f in MC;

FIGS. 5 a and 5 b are views showing a difference of cutting points oncell edges between MC and KTC;

FIG. 6 is a flowchart showing a method and a program for converting dataaccording to the present invention;

FIG. 7 a to 7 f are views showing all equivalence classes of cuttingpoint arrangement in 2D-KTC;

FIGS. 8 a to 8 v are views showing all equivalence classes of cuttingsegment arrangement in the 2D-KTC of FIGS. 7 a to 7 f;

FIG. 9 is a view of a cubic octahedron (one of semiregular polyhedra);

FIGS. 10 a to 10 ad are views showing thirty equivalence classes whenthe number of cutting points is 0 to 4 in cutting point arrangement of3D-KTC;

FIGS. 11 a to 11 aa are views showing twenty seven equivalence classeswhen the number of cutting points is 4 or 5 in the cutting pointarrangement of the 3D-KTC;

FIGS. 12 a to 12 ad are views showing thirty equivalence classes whenthe number of cutting points is 6 in the cutting point arrangement ofthe 3D-KTC;

FIG. 13 is a view showing an example in which a line connecting allgiven cutting points constitutes a closed loop on a surface of a cube;

FIG. 14 is a view representing a relation between 12 vertexes and pathsof 36 edges of the cubic octahedron of FIG. 9 as a graph on a plane;

FIG. 15 is a view showing an example in which no cell division is madeeven if a closed loop is established;

FIGS. 16 a to 16 c are views showing examples in which 2- or 3-divisionof cells can be made depending on a way of triangular division whichuses a closed loop as an edge even if the closed loop is formed;

FIG. 16 d is a view showing a three-dimensional version of FIG. 4 f;

FIG. 17 is a view showing an example in which 2-division of cells ismade even if not all cutting points can be connected by one closed loop;

FIGS. 18 a to 8 c are views showing 14 triangular divisions obtainedfrom Catalan number by assuming a closed loop formed by connectingcutting points as a regular hexagon when the closed loop is a hexagon in3 equivalence classes; and

FIGS. 19 a to 19 d are views showing images on a display which compareB-rep shape representations of a cyclide and a mold with plane formationof KTC.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Next, the preferred embodiments of the present invention will bedescribed with reference to the accompanying drawings.

FIG. 6 is a flowchart of a method and a program for converting dataaccording to the present invention. As shown, each of the method and theprogram of the invention comprises a division step (A), a cutting pointdeciding step (B), a boundary deciding step (C), a cell classificationstep (D), and a boundary cell data classification step (E).

External data 12 input from the outside is polygon data indicating apolyhedron, a tetrahedron or hexahedron element used for a finiteelement method, curved surface data used for three-dimensional CAD or aCG tool, or data in which the other solid surface is represented byinformation containing a partial plane or curved surface.

In addition to such data (referred to as S-CAD data), the external data12 may be (1) data directly prepared by a human input through a boundarysurface of V-CAD's own (V-boundary surface), (2) surface digitized dataof a measuring device, a sensor, a digitizer or the like, and (3) volumedata containing internal information such as voxel data for CT scanning,MRI, and generally used for volume rendering.

In the division step (A), the external data 12 constituted of boundarydata of an object obtained in an external data acquisition step (notshown) is divided into cells 13 of an orthogonal grid.

In the case of a two-dimensional cell, the data is divided intorectangular cells by quad-tree division.

In the case of a three-dimensional cell, the data is divided intorectangular parallelepiped cells 13 by oct-tree division. In oct-treerepresentation, i.e., oct-tree space division, a reference rectangularparallelepiped 13 including a target solid (object) is divided into 8parts, and 8-division processing is recursively repeated to a specifiedcell size until no more boundary surfaces are included in each region.By this oct-tree division, it is possible to reduce an amount of datamore greatly than that in voxel representation.

In the quad-tree representation, one space region obtained by oct-treespace division is called a cell 13. The cell is rectangular orrectangular parallelepiped. The rectangle or rectangular parallelepipedis a square or a cube in a special case, but more generally it may be aquadrangle or a hexahedron in which edges do not cross each other at aright angle. A region occupying a space is represented by a hierarchicalstructure, the number of divisions or resolution of the cell.Accordingly, the object is represented by stacking cells different fromone another in size in the entire space.

In the cutting point deciding step (B), an intersection point ofboundary data and a cell edge is decided as a cell edge cutting point15.

In the case of the two-dimensional cell, in the cutting point decidingstep (B), intersection point patterns 2⁴=16 of boundary data and celledges are set as cell edge cutting point arrangement, and intersectionpoint patterns (arrangement cases or sorts) that become equivalenceclasses by rotational operation are decided as identical patterns sothat the totally 2⁴=16 arrangement cases are further classified into 6patterns.

In the case of the three-dimensional cell, in the cutting point decidingstep (B), intersection point patterns 2¹²=4096 of boundary data and celledges are set as a cell edge cutting point arrangement, and theintersection point patterns (the arrangement cases) that becomeequivalence classes by rotational operation and mirroring operation aredecided as identical patterns so that the totally 2¹²=4096 arrangementcases are further classified into 144 patterns. In the case of thethree-dimensional cell, since there are many patterns, it is furtherpossible to reproduce 144 patterns of 7 to 12 cutting points from 87patterns of 0 to 6 cell edge cutting points by deciding as identicalpatterns the intersection point patterns that become equivalence classesby an inversion operation regarding presence/nonpresence of cuttingpoints.

In the boundary deciding step (C), a boundary formed by connecting theobtained cell edge cutting points is decided as the cell inner shapedata.

In the case of the two-dimensional cell, in the boundary deciding step(C), cutting segment arrangement made by connecting the cell edgecutting points is decided as the cell inner shape data for all the 6patterns of the cell edge cutting points. More specifically, it isadvised to classify the cell inner shape data into 22 kinds of cellinner shape patterns (described later) by deciding as identical patternsthe cell inner shape data that become equivalence classes by athree-dimensional rotational operation.

In the case of the three-dimensional cell, in the boundary deciding step(C), an approximate boundary surface (=cut triangle arrangement) made byconnecting the cell edge cutting points is decided as the cell innershape data for all the patterns of the cell edge cutting points.Incidentally, in a later-described specific example, acquisition of theapproximate boundary surface made by connecting the cell edge cuttingpoints may be referred to as “plane formation”.

In the cell classification step (D), the cells obtained by the divisionare classified into a nonboundary cell 13 a including no boundarysurface and a boundary cell 13 b including a boundary surface.

That is, according to the invention, quad-tree or oct-tree division isused to represent the boundary cell 13 b, a cell completely includedinside is decided as an internal cell 13 a having a largest size, and acell including boundary information from the external data 12 is decidedas a boundary cell 13 b.

In the boundary cell data classification step (E), cell dataconstituting a boundary cell are classified into internal cell data andexternal cell data. The internal cell data is inside the cell innershape data, and the external cell data is outside the cell inner shapedata.

According to the method of the invention, the steps (A) to (E) arerepeated when necessary. Additionally, for example, simulations ofdesigning, analyzing, fabricating, assembling, testing and the like aresequentially carried out by using the obtained V-CAD data 14, and theresult is output as CAM or polygon data, for example at an output step.

The conversion program of the invention is a computer program forexecuting the aforementioned steps (A) to (E), and incorporated in acomputer to be used.

Next, the present invention will be described more in detail.

1. The invention provides a novel method for generating a cell innersurface. This completely includes cell inner surface patterns of amarching cube method, and is a more general method.

According to the invention, when a shape such as a polygon is read intoa voxel aggregation to generate volume data, an intersection point of acell edge and a shape surface which is a sampling point is recorded as acutting point, and the shape surface is restored based on such cuttingpoint information.

According to the invention, to begin with, cell inner surface generationpatterns from the cutting points are completely defined intwo-dimensional representation. That is, enumeration (6 patterns) ofequivalence classes of cutting point arrangement is performed, and allpossibilities (22 patterns) of cell inner surface formation arecollected for all arrangement of cutting points.

For three-dimensional representation, equivalence classes are enumerated(144 patterns) for the cutting point arrangement according to theinvention. Regarding three-dimensional cell inner surface generation,cutting points can be connected on 6 surfaces of a cube by line segmentswithout intersection for more than half of all the arranged cuttingpoints. If such a closed loop is constructed, it is possible torelatively easily form a cell inner surface in which the loop is anouter edge. As a result of the enumeration, arrangement of cuttingpoints which has one closed loop passing through all the cutting pointsonce corresponds to 87 patterns among the 144 patterns. An example of acell inner surface formed by using a closed loop will be describedlater.

1.1 It is a theme of the invention that a surface shape of conventionaltype CAD is read into a cell space, an intersection point (cuttingpoint) of a surface and a cell edge is recorded, and the originalsurface shape is accordingly approximated to a triangular surface.

1.2 Marching Cubes (MC)

In the case of the aforementioned marching cubes (abbreviated to MC),positive and negative numerical values are written in 8 vertexes of thecell, and isosurfaces are generated based on the positive and negativenumerical values (isosurfaces of zero values are consideredhereinafter). If signs (positive or negative) of the numerical values ofboth ends of an edge of a cube are different from each other, onecutting point is disposed on the edge. No cutting point is disposed ifsigns of the numerical values are equal to each other. This operation iscarried out for 12 edges of the cube, and then a surface is formed basedon the cutting points.

The MC has come into wide use since the surface is strongly formed by asimple algorithm. Contents of the MC can be summarized as the followingtwo: (1) pattern classification of cutting point arrangement, and (2)definition of formation of a cell inner surface for each cutting pointarrangement (i.e., how to connect cutting points). A problem ofambiguity of the MC is that formation of a cell inner surface of (2) isnot always unique. However, it is not difficult to specify uniqueformation of a surface. For example, if ambiguity arises, it is advisedto decide formation of a cell inner surface to interconnect vertexes oflarge values, or vertexes of small values. The strong MC means that thecell inner surface can be formed without holes by such decision, i.e., atriangular edge constituting the cell inner surface is shared by 2triangles.

1.3 Kitta Cubes (KTC)

The invention provides novel broad-sense means for approximating asurface shape by a cut triangle in which a cutting point is a vertex,which is referred to as Kitta cubes (abbreviated to KTC hereinafter)(Kitta is a Japanese word meaning “cut”).

In the case of the aforementioned MC, the cell inner surface is stronglyformed. However, it does not necessarily mean that the shape iscorrectly represented. The invention makes it apparent that when theshape is represented at equal resolution (=equal cell sizes), it can berepresented more accurately by using the KTC than by using the MC.

Before describing accurate definition of the KTC, a two-dimensionaltypical example showing that the KTC can represent the shape moreaccurately than the MC is given (each of FIGS. 4 a to 4 c). In the caseof the aforementioned MC, in a situation as shown in FIG. 4 a, 4vertexes of a square cell have the same sign as they are located outsidethe shape, and thus no cutting points are disposed on 4 edges of thecell. As a result, no cell inner surface is formed at all in the case ofFIG. 4 a. From the standpoint of the MC, this means that currentresolution is too rough to represent the shape.

Next, the KTC is applied at equal resolution. Cutting points as shown inFIG. 4 b are recorded by cutting point checking. If the number ofcutting points on one edge is up to one, and two cutting points of FIG.4 b are integrated into one in a proper position, a cell inner surfaceas shown in FIG. 4 c is formed at the end. A difference in power ofrepresentation between the two is apparent. FIG. 4 a shows an extremeexample. However, essentially similar defects frequently occur, and FIG.4 d shows such an example. The defects frequently occur at anintersection of a curved boundary surface and a cell edge. In the caseof the KTC, this situation is approximated to that of FIG. 4 f. In thecase of the MC, it is approximated to that of FIG. 4 g. FIGS. 4 c and 4f show two-dimensional examples. In three-dimensional representation,more cases can be represented only by the KTC. It can be understood thatthe KTC is much higher in power of representation than the MC at equalresolution.

In fact, if algorithm frameworks are compared between the KTC and theMC, the KTC completely includes the MC (described later). That is, MCpatterns constitute a partial aggregation of KTC patterns. In thissense, it is only natural that the KTC is higher in power ofrepresentation than the MC.

In each of FIGS. 4 a to 4 c, the number of cutting points on one edge islimited to one. However, to understand the frameworks of the MC and theKTC, it is important to recognize that this limitation has an apparentdifference in specific meaning. In MC construction, the number ofcutting points is limited to 0 or 1. According to the invention, the KTCis constructed within a framework in which the number of cutting pointsis 0 or 1 on one edge. A difference between the two is that one cuttingpoint is disposed on one edge only when signs of both ends of each edgeof the cell oppose each other (positive and negative) (FIG. 5 a) in thecase of the MC, while a cutting point is disposed on one edge not onlywhen signs of both ends thereof oppose each other (positive andnegative) but also when signs of the both ends thereof are equal to eachother in the case of the KTC (FIG. 5 b).

Hereinafter, the case of the MC is referred to as “narrow-sensecondition that the number of cutting points on one edge is at most one”,and the case of the KTC is referred to as “broad-sense condition thatthe number of cutting points on one edge is at most one” to clearlydistinguish them from each other. As apparent from FIGS. 5 a and 5 b,the broad-sense condition that the number of cutting points on one edgeis at most one completely include the narrow-sense condition that thenumber of cutting points on one edge is at most one. Thus, regardingpatterns of two-dimensional and three-dimensional cutting pointarrangements provided by both, all the MC patterns constitute a partialaggregation of all the KTC patterns.

2. Two-dimensional Kitta cubes KTC (2D-KTC) is constructed intwo-dimensional representation.

2. 1 Classification of Cutting Point Arrangement in 2D-KTC

The 2D-KTC is constructed at 2 stages. The first stage is classificationof cutting point arrangement for 4 edges of a square. The second stageis classification of cutting segment (=segment having cutting points inboth ends thereof) arrangement for each cutting point arrangement.

FIGS. 7 a to 7 f show all the patterns of the cutting point arrangementin the 2D-KTC. There are 2⁴32 16 patterns because of two possibilitiesof presence and nonpresence of cutting points for each of the 4 edges.These are integrated into 6 patterns of Table 1 and FIGS. 7 a to 7 f ifequivalence classes are enumerated for a rotational operation. In thedrawings, circled squares are present in the 2D-MC.

If an inversion operation regarding presence/nonpresence of cuttingpoints is added to classification of equivalence classes, the 6 patternsare integrated into 4 patterns. However, in cutting segment arrangementof the section 2. 2, all the 6 patterns must be defined.

TABLE 1 Number of cutting points: 1, 3 2 Total(K = 0-4) Total number 4 616 of cutting point arrangements: [4k] Number of 1 2  6 (4) equivalenceclasses Index in FIGS. 7a FIG. 7a, FIG. 7b, FIG. 7a, — to 7f FIG. 7fFIG. 7e FIG. 7f

In appendices A, B (described later), 2D and 3D MC are described. In theMC, black and white inversion of black and white coloring of vertexes(corresponding to distinction between positive and negative values ofthe vertexes) is permitted in classification. It is because of no changein cutting point arrangement before and after black and white inversion.

Degrees of reduction for the arrangements in FIGS. 7 a to 7 f are 1, 4,4, 2, 4, 1 in order (16 if added together). Here, the “degree ofreduction” means the number of arrangements integrated into oneequivalence class by a rotational operation.

2. 2. Cutting Segment Arrangement in 2D-KTC

The process proceeds to the second stage of the 2D-KTC. Here, all thepossibilities of cutting segment arrangements are collected for each ofthe 6 kinds of cutting point arrangements obtained in the section 2. 1.In the collection, when cutting segments are arranged in a target cell,it is not necessary to use all the cutting points present on the edge ofthe cell”. One edge is shared by two cells. If there is one cuttingpoint on the edge, even when the edge is not used in one cell to form acell inner line, there is a possibility that it will be used in theother cell.

Needless to say, there is a possibility that the edge will be used inneither of the cells. It is when a very small shape intersects only oneedge. An attitude of utilizing cutting point information is to place avery small segment on the edge, and the other attitude is not to use thecutting points for line setting at all. However, since the placing ofthe segment (placing of surface part in 3 dimensional representation)means that a shape surface becomes other than a manifold, it is notdescribed in this specification. For the same reason, a case in whichcutting segments intersect each other in a cell or branched (=threecutting segments intersect one another on the same cell edge) isexcluded from enumeration.

FIGS. 8 a to 8 v show all arrangements of cutting segments in the2D-KTC, which are results of investigating all possibilities of linesetting for each of the 6 kinds of cutting point arrangement obtained inthe section 2. 1. In the drawings, circled squares are present in the2D-MC. These apparently constitute only a partial aggregation of the2D-KTC.

A three-dimensional rotational operation is used when equivalenceclasses are counted. Assuming that connection of cutting segments haveno diversity, i.e., intersections or branches, it is possible todifferently color 4 vertexes of FIGS. 8 a to 8 v black and white.Different from the case of the MC, the different coloring is executedafter arrangement of the cutting segments. Paying attention to cuttingsegment arrangement patters of FIGS. 8 a to 8 v, similar types emerge asin the case of, e.g., 2A(1), 3(5), 4(9). However, they cannot be omittedfor the purpose of including ways of surface formation for all thecutting point arrangements. Incidentally, there are 10 cases if 22 kindsof patterns drawn in FIGS. 8 a to 8 v are classified by cutting segmentarrangement.

3. Three-dimensional Kitta Cubes (3D-KTC)

3. 1 Classification of Cutting Point Arrangement in 3D-KTC

It is made clear how many equivalence classes totally 2¹²=4096 cut edgearrangements are integrated into and how they are arranged when only arotational operation and a mirroring operation are permitted.

To begin with, since equations (1) and (2) of an [expression 1] areestablished, it can be understood that an inversion operation regardingpresence of cutting points in arrangement in which the number of cuttingpoints is 0 to 5 is carried out to obtain arrangement in which thenumber of cutting points is 7 to 12, and correspondence is set forequivalence classes by the inversion operation.

[Expression 1]

$\begin{matrix}{2^{12} = {\sum\limits_{k = 0}^{12}\begin{pmatrix}12 \\k\end{pmatrix}}} & (1) \\{\begin{pmatrix}12 \\k\end{pmatrix} = \begin{pmatrix}12 \\{12 - k}\end{pmatrix}} & (2)\end{matrix}$

Indexes of 1 to 12 are first given for sides of a cube as shown in FIG.9. 48 substitution operations are enabled by adding a mirroringoperation to 24 rotational operations of a group of regular octahedra.Checking is made on whether the indexes of the 4096 arrangements are newor old one by one in order from the first. In this case, past appearanceof the same arrangements can be determined by executing 48 operationsfor individual arrangements and collating them with the existing list.As shown in Table 2, 144 kinds of cutting point arrangements differentfrom one another can be obtained at the end.

TABLE 2 Number of cutting points: k Total 0, 12 1, 11 2, 10 3, 9 4, 8 5,7 6 (K = 0-12) Total number of 1 12 66 220 495 792 924 4096     cuttingpoint arrangements [12k] Number of equivalence 1  1  4  9  18  24  30 144 (87) classes

As described in the previous paragraph, the equivalence classes in whichthe number of cutting points is 7 to 12 can be reproduced from theequivalence classes in which the number of cutting points is 0 to 5.Thus, totally 87 patterns in which the number of cutting points is 0 to6 are collected in Table 3, and shown in FIGS. 10 a to 10 ad, FIGS. 11 ato 11 aa, and FIGS. 12 a to 12 ad. Circled cubes are present in the3D-MC. It can be understood that cutting point arrangement cases of theMC constitute only a part of KTC cutting point arrangement cases. InFIGS. 10 a to 10 ad, FIGS. 11 a to 11 aa, and FIGS. 12 a to 12 ad, acase in which the number of cutting points is 7 to 12 is omitted.However, as in the case of the 2D-KTC, 144 patterns are necessary whenarrangement of cut triangles is considered.

TABLE 3

0 cutting point 0-1 None

1 cutting point 1-1 (1)

2 cutting points 2-1 (1 2) 2-2 (1 3) 2-3 (1 6) 2-4 (1 7)

3 cutting points 3-1 (1 2 3) 3-2 (1 2 5) 3-3 (1 2 7) 3-4 (1 2 9) 3-5 (13 6) 3-6 (1 3 9) 3-7 (1 3 10) 3-8 (1 3 12) 3-9 (1 7 12)

4 cutting points 4-1 (1 2 3 4) 4-2 (1 2 3 5) 4-3 (1 2 3 7) 4-4 (1 2 3 8)4-5 (1 2 3 9) 4-6 (1 2 3 10) 4-7 (1 2 5 6) 4-8 (1 2 5 7) 4-9 (1 2 7 8)4-10 (1 2 7 9) 4-11 (1 2 7 10) 4-12 (1 2 9 11) 4-13 (1 2 9 12) 4-14 (1 36 8) 4-15 (1 3 6 9) 4-16 (1 3 6 10) 4-17 (1 3 6 11) 4-18 (1 3 6 12)

5 cutting points 5-1 (1 2 3 4 5 10) 5-2 (1 2 3 4 9) 5-3 (1 2 3 5 6) 5-4(1 2 3 5 7) 5-5 (1 2 3 5 8) 5-6 (1 2 3 5 9) 5-7 (1 2 3 5 10) 5-8 (1 2 35 11) 5-9 (1 2 3 5 12) 5-10 (1 2 3 7 8) 5-11 (1 2 3 7 9) 5-12 (1 2 3 710) 5-13 (1 2 3 8 9) 5-14 (1 2 3 8 10) 5-15 (1 2 3 9 11) 5-16 (1 2 3 912) 5-17 (1 2 3 10 12) 5-18 (1 2 5 7 11) 5-19 (1 2 5 7 12) 5-20 (1 2 7 89) 5-21 (1 2 7 9 12) 5-22 (1 2 7 10 12) 5-23 (1 3 6 8 9) 5-24 (1 3 6 810) — —

6 cutting points 6-1 (1 2 3 4 5 6) 6-2 (1 2 3 4 5 7) 6-3 (1 2 3 4 5 9)6-4 (1 2 3 4 5 11) 6-5 (1 2 3 4 9 10) 6-6 (1 2 3 4 9 12) 6-7 (1 2 3 5 67) 6-8 (1 2 3 5 6 8) 6-9 (1 2 3 5 6 9) 6-10 (1 2 3 5 6 10) 6-11 (1 2 3 57 8) 6-12 (1 2 3 5 7 9) 6-13 (1 2 3 5 7 10) 6-14 (1 2 3 5 7 11) 6-15 (12 3 5 7 12) 6-16 (1 2 3 5 8 9) 6-17 (1 2 3 5 8 10) 6-18 (1 2 3 5 8 11)6-19 (1 2 3 5 8 12) 6-20 (1 2 3 5 11 12) 6-21 (1 2 3 7 8 10) 6-22 (1 2 37 9 10) 6-23 (1 2 3 7 9 12) 6-24 (1 2 3 7 10 12) 6-25 (1 2 3 8 9 11)6-26 (1 2 3 8 9 12) 6-27 (1 2 3 8 10 12) 6-28 (1 2 7 8 9 12) 6-29 (1 3 68 9 12) 6-30 (1 3 6 8 10 11) — —

The 144 kinds of arrangements can all emerge as cutting pointarrangements when a surface shape is actually read into a cell space.Table 7 of the appendix B shows correspondence of all the cutting pointarrangements between the 3D-MC and the 3D-KTC. It can be understood thatthe MC arrangements constitute only a part of the KTC arrangements.

3. 2 Regarding Definition of Surface Formation in 3D-KTC

FIG. 13 shows an example in which segments made by connecting all givencutting points constitutes a closed loop on a surface of a cell. It is acubic octahedron (one of semiregular polyhedra. 14-hedron constituted of8 regular triangles and 6 squares) that is shown together with the cellin FIG. 9. Even if a position of a cutting point is a center point ofeach edge, generality is not lost when cutting point arrangements arecounted, or a closed loop is considered. This cubic octahedron is aconvex hull for 12 center points on 12 edges. As paths for tracing theclosed loop, there are totally 36 paths constituted of two diagonallines of each square surface of the 14-hedron, i.e., 12 diagonal lines,and 24 edges which the 14-hedron originally has.

FIG. 14 shows a relation between 12 vertexes of the semiregularpolyhedron and paths of 36 edges as a graph on a plane. Enumeration ofHamilton closed paths will be described with reference to this drawing.In this graph, the 12 vertexes are all equal at values of 6, andenumeration here needs to be carried out only from one vertex. Pathspermitted as Hamilton closed paths are those which pass through allcutting points given on the surface of the cube so as to return to thestart point without passing through the same path or cutting point againin the midway. A result of the enumeration is 87 kinds which constitutemore than half of the total 144 kinds of the cutting point arrangements,the number of cutting points being 0 to 12 therein, and havenon-self-intersection Hamilton paths.

In fact, many of cell inner surfaces formed by such closed loops aresuited to V-CAD. The V-CAD has physical information as cell internalinformation as described above, and multimedia can be dealt with. Inthis case, a plurality of substances occupy the inside of the cell on aboundary surface of a different material. When the number of substancesbecomes 3 kinds or more, the number of substances in one cell ispreferably limited to two by executing given processing. Many of cellinner surfaces formed by the closed loops are advantageous in that thecell is divided into two.

However, all the closed loops do not necessarily divide the cell intotwo. For example, the cell may not be divided even if a closed loop isformed as shown in FIG. 15. Examples of FIGS. 16 a to 16 c show that thecell can be divided into two or three depending on a way of triangulardivision in which the closed loop is set as an edge even if the closedloop is formed. Conversely, the cell may be divided into two even if oneclosed loop passing through all the cutting points is not formed. FIG.17 shows such an example.

3. 3 Regarding Triangular Division

If the closed loop is constructed, it is easy in many cases to form acell inner surface in which the closed loop is set as an edge. A problemof triangularly dividing the cell inner surface has a relation to aproblem of triangular division of a polygon on a plane. It is known thatthe number of triangular division patterns is represented by usingCatalan numbers. The Catalan numbers and the triangular division aredescribed later in the appendix C.

3. 4 Example of 3D-KTC

Execution using a closed loop was tried. B-rep shape representation wasread to record cutting points, and a closed loop was searched thereforto carry out triangular division. Examples are a cyclide (FIG. 19 a) anda mold (FIG. 19 b). FIGS. 19 c and 19 d show restoration thereof basedon cutting point information. These are generated from closed loopswhich are only parts of KTC surface formation possibility, but they havesufficient power of representation.

It must be emphasized that the cutting point arrangements correspondingto the closed loops are only parts of the KTC, but they do not fallwithin the cutting point arrangement framework of the MC. While thereare only totally 14 cutting point arrangements in the MC, it is 87 kindsamong the 144 cutting point arrangements of the KTC that correspond tothe closed loops as described above. As an example, the cutting pointarrangements of FIGS. 16 a to 16 c can constitute closed loops, but theyare patterns nonpresent in the MC. Further, a three-dimensional versionof FIG. 4 f is shown in FIG. 16 d.

4. Conclusion

As described above, the KTC is capable of more precise shaperepresentation than the MC when they are compared at equal resolution.This is only natural because the equivalence classes of the cuttingpoint triangular arrangement of the KTC completely include those of theMC.

The present invention precisely presents the diversity exhibited by theKTC on the assumption of only the broad-sense condition that the numberof cutting points on one edge is at most one.

Appendix A: Two-dimensional Marching Cubes (2D-MC)

FIGS. 2 a to 2d show 4 kinds of equivalence classes of the 2D-MC. These4 patterns are obtained by using only a black and white inversionoperation and a three-dimensional rotational operation on 4 vertexes ofa cell. Since the MC abides by the narrow-sense condition that thenumber of cutting points on one edge is at most, 4 vertex patternsdifferently colored black and white and cutting point arrangement casescorrespond to each other by 1 to 1. In FIGS. 2 a to 2 d, indexes ofmc-0, mc-1, mc-2, and mc-3 are allocated to patterns.

The 2D-KTC is described in the section 2.1. It should be noted that thetotal numbers of arrangements before classification of the equivalenceclasses are equally 16 (=2⁴) in the MC and the KTC in two-dimensionalrepresentation. This is attributed to the fact that the numbers ofvertexes and sides of a square are both 4. (In the case of a cube, sincethe numbers of vertexes and edges are different from each other, thetotal number of cutting point arrangements is 4096 (=2¹²) in the 3D-KTCwhile it is 256 (=2⁸) in the 3D-MC.

Table 4 shows correspondence between the number of equivalence classesregarding cutting point arrangement, and the total number ofarrangements. Table 5 shows correspondence between the cutting pointarrangement of the MC and the cutting point arrangement of the KTC. Thecutting point arrangements of the KTC that do not appear here, i.e.,those of FIGS. 7 a and 7 e, are not present in the cutting pointarrangement of the MC.

TABLE 4 Number of black vertexes k 0(or 4) 1(or 3) 2 Total(k = 0-4)Total number of 1 4 6 16 arrangements [4k] Number of 1 1 2 4 equivalenceclasses

TABLE 5 2D-MC index in FIGS. 2a to 2d 0 1 2 3 2D-KTC index in FIGS. 8ato 8v 0(1) 2A(1) 2B(1) 4(4)

Appendix B: Three-dimensional Marching Cubes (3D-MC)

FIGS. 3 a to 3 n show all the equivalence classes of the MC cuttingpoint arrangement. Operations used for classification are a black andwhite inversion operation, a rotational operation and a mirroringoperation on vertexes. As a result, vertex coloring of 256 (=2⁸) kindsis classified into 14 patterns. Table 6 shows an aggregation of theseregarding the number of cutting points.

TABLE 6 Number of black vertexes: k 0 1 2 3 Total (or 8) (or 7) (or 6)(or 5) 4 (k = 0-8) Total number of 1 8 28 56 70 256 black vertexarrangements: [8k] Number of 1 1 3 3 6 14 equivalence classes

It should be noted that the MC is represented by totally 15 drawings of0 to 14 in accordance with a normal MC original thesis, but only 14 areshown here (FIGS. 3 a to 3 n). The omitted MC-14 is reflectionalsymmetry of the MC-11. Since the MC-15 is uniquely decided by amirroring operation after surface formation of the MC-11 is decided, itis omitted to maintain consistency of description of the invention (useof mirroring operation for enumeration of equivalence classes).

The FIGS. 3 a to 3 b are similar in lining-up to drawings of theoriginal thesis, but a cell inner surface is omitted, and the MC-14 isomitted for the aforementioned reason. Table 7 shows correspondence ofall the cases between the MC and the KTC.

TABLE 7 3D-MC index in FIGS. 3a to 3N 0 1 2 3 4 5 6 7 8 9 10 11 12 133D-KTC 0-1 3-6 4-12 6-6 6-29 5-18 7-18 9-6 4-7 6-30 8-7 6-28 8-12 12-1index in FIGS. 10a to 10ad, FIGS. 11a to 11aa, FIGS. 12a to 12ad

Appendix C: Triangular division and Catalan number of polygon

A problem of a folded surface in which a closed loop tracing a cellsurface is an edge, i.e., triangular division of the inside of the loop,can come down to a problem of triangularly dividing a convex polygonshape on a plane in many cases. (If the entire closed loop is notplanar, i.e., if a greatly bent cell inner surface is formed by theclosed loop, the problem may not come down to the latter problem).

A problem of obtaining a number in the case of triangular division isknown to have a relation to Catalan numbers. Only a result is describedbelow.

Catalan number C_(m) is represented by an equation (3) of an expression(2). In this case, when the convex n square shape (convex polygon shape)is triangularly divided by diagonal lines which do not cross each othertherein, how many different ways of division are available is obtained.An obtained number T_(n) is represented by an equation (4) of theexpression (2).

[Expression 2]

$\begin{matrix}{C_{n} = {\frac{1}{m + 1}\begin{pmatrix}{2m} \\m\end{pmatrix}}} & (3) \\{T_{n} = {C_{n - 2} = {\frac{1}{n - 1}\begin{pmatrix}{2( {n - 2} )} \\{n - 2}\end{pmatrix}}}} & (4)\end{matrix}$

If n=3, 4, . . . , 12 are substituted for the equation, T₃=1, T₄=2,T₅=5, T₆=14, . . . , T₁₂=16796 are obtained.

Calculation is not so difficult at n=5 or lower. Supplementalexplanation is made of T₆=14. 14 are integrated into 3 patterns shown inFIGS. 18 a to 18 c by triangle rotation for a regular hexagon. FIGS. 18a to 18 c show representative equivalence classes of 6, 2 and 6triangular division methods. It can be confirmed that these are added tobe 14.

As described above, according to the method and the program of thepresent invention, by the division step (A) and the cell classificationstep (D), it is possible to store the external data (12) of the objectas a cell hierarchy in which the external cell (12) is divided into theorthogonal cells (13) by a small storage capacity.

In the cutting point deciding step (B), the intersection point of theboundary data and the cell edge is decided as the cell edge cuttingpoint. Thus, it is possible to include arrangements of all the cuttingpoints by the MC in which “one cutting point is disposed on an edge ifsigns of numerical values of both ends of the cell edge are differentfrom each other, and no cutting point is disposed if signs of thenumerical values are equal to each other”, and to include arrangementsof all the cutting points on the boundary surfaces and the cell edgelines without omission under the condition that the number of cuttingpoints on one edge is at most one.

Further, in the boundary deciding step (C), the boundary formed byconnecting the obtained cell edge cutting points is decided as the cellinner shape data. Thus, it is possible to include all the cell innershape patterns by the MC, and to include the cell inner cut trianglearrangements without omission under the condition that the number ofcutting points on one edge is at most one.

Additionally, in the boundary cell data classification step (E), thecell data constituting the boundary cell are classified into theinternal cell data inside the cell inner shape data and the externalcell data outside the cell inner shape data. Thus, it is possible toclassify all the cell data into nonboundary and boundary cell data whilemaintaining continuity from the adjacent cells.

Therefore, the method and the program of the invention for convertingthe boundary data into the cell inner shape data are advantageous inthat it is possible to include all the cell edge cutting points by theMC and all the other cell edge cutting points without omission under thecondition that the number of cutting points on one edge is at most one,and it is thereby possible to include the cell inner shape dataconstituted of the boundary connecting the cell edge cutting pointswithout omission.

The invention has been described by way of preferred embodiments.However, it can be understood that a scope of claims of the invention isnot limited to the embodiments. As a matter of fact, all kinds ofimprovements, modifications and equivalents are within the scope of theappended claims of the invention.

1. A method for converting boundary data into cell inner shape data,comprising: a division step (A) of dividing external data constituted ofthe boundary data of an object into cells in an orthogonal grid; acutting point deciding step (B) of deciding an intersection point of theboundary data and a cell edge as a cell edge cutting point; a boundarydeciding step (C) of deciding a boundary formed by connecting the celledge cutting points as the cell inner shape data; a cell classificationstep (D) of classifying the divided cells into a nonboundary cellincluding no boundary surface and a boundary cell including a boundarysurface; a boundary cell data classification step (E) of classifyingcell data constituting the boundary cell into internal cell data insidethe cell inner shape data and external cell data outside the cell innershape data; and step (F) of outputting the cell inner shape data to adisplay, wherein the cells are rectangular cells in two-dimensionalrepresentation, and in the cutting point deciding step (B), intersectionpoints of boundary data and cell edges that have totally 2⁴=16arrangement cases are decided as the cell edge cutting points, and thearrangement cases that become equivalence classes by rotationaloperation are decided as identical patterns so that the 2⁴=16arrangement cases are further classified into 6 patterns, and in thecutting point deciding step (B), as for arrangement of each intersectionpoint, data concerning the rotational operation and data concerning theidentical pattern are stored in a storage device, wherein in theboundary deciding step (C), a boundary line made by connecting the celledge cutting points is decided as the cell inner shape data for all the6 patterns.
 2. The method according to claim 1, characterized in that:in the boundary deciding step (C), cell inner shape data patterns thatbecome equivalence classes by three-dimensional rotational operation aredecided as identical patterns so that the cell inner shape data patternsare classified into 22 patterns.
 3. The method according to claim 1,further comprising the step of: showing images on the display using thecell inner shape data.
 4. A computer readable medium encoded with aprogram for converting boundary data into cell inner shape data, whereinthe program causes a computer to execute: a division step (A) ofdividing external data constituted of boundary data of an object intocells in an orthogonal grid; a cutting point deciding step (B) ofdeciding an intersection point of the boundary data and a cell edge as acell edge cutting point; a boundary deciding step (C) of deciding aboundary connecting formed by the cell edge cutting points as the cellinner shape data; a cell classification step (D) of classifying thedivided cells into a nonboundary cell including no boundary surface anda boundary cell including a boundary surface; a boundary cell dataclassification step (E) of classifying cell data constituting theboundary cell into internal cell data inside the cell inner shape dataand external cell data outside the cell inner shape data; and step (F)of outputting the cell inner shape data to a display, wherein the cellsare rectangular cells in two-dimensional representation, and in thecutting point deciding step (B), intersection points of boundary dataand cell edges that have totally 2⁴=16 arrangement cases are decided asthe cell edge cutting points, and the arrangement cases that becomeequivalence classes by rotational operation are decided as identicalpatterns so that the 2⁴=16 arrangement cases are further classified into6 patterns, and in the cutting point deciding step (B), as forarrangement of each intersection point, data concerning the rotationaloperation and data concerning the identical pattern are stored in astorage device, wherein in the boundary deciding step (C), a boundaryline made by connecting the cell edge cutting points is decided as thecell inner shape data for all the 6 patterns.
 5. The program accordingto claim 4, wherein the program further causes the computer to executethe step of: showing images on the display using the cell inner shapedata.
 6. A method for converting boundary data into cell inner shapedata, comprising the steps of: (A) dividing external data constituted ofthe boundary data of an object into cells in an orthogonal grid; (B)deciding an intersection point of the boundary data and a cell edge as acell edge cutting point; (C) deciding a boundary formed by connectingthe cell edge cutting points as the cell inner shape data; (D)classifying the divided cells into a nonboundary cell including noboundary surface and a boundary cell including a boundary surface; (E)classifying cell data constituting the boundary cell into internal celldata inside the cell inner shape data and external cell data outside thecell inner shape data; and (F) outputting the cell inner shape data to adisplay, wherein the cells are rectangular cells in two-dimensionalrepresentation, and in step (B), intersection points of boundary dataand cell edges that have totally 2⁴=16 arrangement cases are decided asthe cell edge cutting points, and the arrangement cases that becomeequivalence classes by rotational operation are decided as identicalpatterns so that the 2⁴=16 arrangement cases are further classified into6 patterns, and in step (B), as for arrangement of each intersectionpoint, data concerning the rotational operation and data concerning theidentical pattern are stored in a storage device.
 7. The methodaccording to claim 6, further comprising the step of: showing images onthe display using the cell inner shape data.
 8. A computer readablemedium encoded with a program for converting boundary data into cellinner shape data, wherein the program causes a computer to execute thesteps comprising: (A) dividing external data constituted of boundarydata of an object into cells in an orthogonal grid; (B) deciding anintersection point of the boundary data and a cell edge as a cell edgecutting point; (C) deciding a boundary connecting formed by the celledge cutting points as the cell inner shape data; (D) classifying thedivided cells into a nonboundary cell including no boundary surface anda boundary cell including a boundary surface; (E) classifying cell dataconstituting the boundary cell into internal cell data inside the cellinner shape data and external cell data outside the cell inner shapedata; and (F) outputting the cell inner shape data to a display, whereinthe cells are rectangular cells in two-dimensional representation, andin step (B), intersection points of boundary data and cell edges thathave totally 2⁴=16 arrangement cases are decided as the cell edgecutting points, and the arrangement cases that become equivalenceclasses by rotational operation are decided as identical patterns sothat the 2⁴=16 arrangement cases are further classified into 6 patterns,and in step (B), as for arrangement of each intersection point, dataconcerning the rotational operation and data concerning the identicalpattern are stored in a storage device.
 9. The program according toclaim 8, wherein the program further causes the computer to execute thestep of: showing images on the display using the cell inner shape data.10. A method for converting boundary data into cell inner shape data,comprising: a division step (A) of dividing external data constituted ofthe boundary data of an object into cells in an orthogonal grid; acutting point deciding step (B) of deciding an intersection point of theboundary data and a cell edge as a cell edge cutting point; a boundarydeciding step (C) of deciding a boundary formed by connecting the celledge cutting points as the cell inner shape data; a cell classificationstep (D) of classifying the divided cells into a nonboundary cellincluding no boundary surface and a boundary cell including a boundarysurface; a boundary cell data classification step (E) of classifyingcell data constituting the boundary cell into internal cell data insidethe cell inner shape data and external cell data outside the cell innershape data; and step (F) of outputting the cell inner shape data to adisplay, wherein the cells are rectangular parallelepiped cells, and inthe cutting point deciding step (B), intersection points of boundarydata and cell edges that have totally 2¹²=4096 arrangement cases aredecided as the cell edge cutting points, and the arrangement cases thatbecome equivalence classes by rotational operation and mirroringoperation are decided as identical patterns so that the 2¹²=4096arrangement cases are further classified into 144 patterns, and in thecutting point deciding step (B), as for arrangement of each intersectionpoint, data concerning the rotational operation and mirroring operationand data concerning the identical pattern are stored in a storagedevice, wherein in the boundary deciding step (C), a boundary line madeby connecting the cell edge cutting points is decided as the cell innershape data for all the 144 patterns.
 11. The method according to claim10, characterized in that: in the cutting point deciding step (B), thecell edge cutting point patterns that become equivalence classes by aninversion operation regarding presence/nonpresence of cutting points aredecided as identical patterns so that the cell edge cutting pointpatterns are classified into 87 patterns in which the number of the celledge cutting points is 0 to
 6. 12. The method according to claim 10,further comprising the step of: showing images on the display using thecell inner shape data.
 13. A method for converting boundary data intocell inner shape data, comprising the steps of: (A) dividing externaldata constituted of the boundary data of an object into cells in anorthogonal grid; (B) deciding an intersection point of the boundary dataand a cell edge as a cell edge cutting point; (C) deciding a boundaryformed by connecting the cell edge cutting points as the cell innershape data; (D) classifying the divided cells into a nonboundary cellincluding no boundary surface and a boundary cell including a boundarysurface; (E) classifying cell data constituting the boundary cell intointernal cell data inside the cell inner shape data and external celldata outside the cell inner shape data; and (F) outputting the cellinner shape data to a display, wherein the cells are rectangularparallelepiped cells, and in step (B), intersection points of boundarydata and cell edges that have totally 2¹²=4096 arrangement cases aredecided as the cell edge cutting points, and the arrangement cases thatbecome equivalence classes by rotational operation and mirroringoperation are decided as identical patterns so that the 2¹²=4096arrangement cases are further classified into 144 patterns, and in step(B), as for arrangement of each intersection point, data concerning therotational operation and mirroring operation and data concerning theidentical pattern are stored in a storage device.
 14. The methodaccording to claim 13, further comprising the step of: showing images onthe display using the cell inner shape data.
 15. A computer readablemedium encoded with a program for converting boundary data into cellinner shape data, wherein the program causes a computer to execute: adivision step (A) of dividing external data constituted of the boundarydata of an object into cells in an orthogonal grid; a cutting pointdeciding step (B) of deciding an intersection point of the boundary dataand a cell edge as a cell edge cutting point; a boundary deciding step(C) of deciding a boundary formed by connecting the cell edge cuttingpoints as the cell inner shape data; a cell classification step (D) ofclassifying the divided cells into a nonboundary cell including noboundary surface and a boundary cell including a boundary surface; aboundary cell data classification step (E) of classifying cell dataconstituting the boundary cell into internal cell data inside the cellinner shape data and external cell data outside the cell inner shapedata; and step (F) of outputting the cell inner shape data to a display,wherein the cells are rectangular parallelepiped cells, and in thecutting point deciding step (B), intersection points of boundary dataand cell edges that have totally 2¹²=4096 arrangement cases are decidedas the cell edge cutting points, and the arrangement cases that becomeequivalence classes by rotational operation and mirroring operation aredecided as identical patterns so that the 2¹²=4096 arrangement cases arefurther classified into 144 patterns, and in the cutting point decidingstep (B), as for arrangement of each intersection point, data concerningthe rotational operation and mirroring operation and data concerning theidentical pattern are stored in a storage device, wherein in theboundary deciding step (C), a boundary line made by connecting the celledge cutting points is decided as the cell inner shape data for all the144 patterns.
 16. The program according to claim 15, wherein the programfurther causes the computer to execute the step of: showing images onthe display using the cell inner shape data.
 17. A computer readablemedium encoded with a program for converting boundary data into cellinner shape data, wherein the program causes a computer to execute thesteps comprising: (A) dividing external data constituted of the boundarydata of an object into cells in an orthogonal grid; (B) deciding anintersection point of the boundary data and a cell edge as a cell edgecutting point; (C) deciding a boundary formed by connecting the celledge cutting points as the cell inner shape data; (D) classifying thedivided cells into a nonboundary cell including no boundary surface anda boundary cell including a boundary surface; (E) classifying cell dataconstituting the boundary cell into internal cell data inside the cellinner shape data and external cell data outside the cell inner shapedata; and (F) outputting the cell inner shape data to a display, whereinthe cells are rectangular parallelepiped cells, and in step (B),intersection points of boundary data and cell edges that have totally2¹²=4096 arrangement cases are decided as the cell edge cutting points,and the arrangement cases that become equivalence classes by rotationaloperation and mirroring operation are decided as identical patterns sothat the 2¹²=4096 arrangement cases are further classified into 144patterns, and in step (B), as for arrangement of each intersectionpoint, data concerning the rotational operation and mirroring operationand data concerning the identical pattern are stored in a storagedevice.
 18. The program according to claim 17, wherein the programfurther causes the computer to execute the step of: showing images onthe display using the cell inner shape data.